Those categories are different from the ones used in many Arabic classes in the West, where sentences are classified simply according to whether or not they include a verb — regardless of where the verb is in the sentence.

Equational sentence - a sentence without a verb.

الولد مصري. (al-walad miSri) - The boy is Egyptian.
Although this sentence contains a verb in English, it doesn't in Arabic. Remember that since Arabic doesn't use a present-tense form of "to be," this is a verbless sentence consisting only of a noun and adjective. It literally translates as "The boy Egyptian"; the "is" is understood.

في أسرتي ثلاثة أفراد. (fi usrati talaatat afraad) - There are three people in my family.
Again, the English translation contains a verb ("are"), but the original Arabic doesn't. It literally translates as "In my family three individuals." Since there's no present-tense form of "to be," "In my family [are] three individuals" is implied.

Subject and predicate

The subject (المبتداء al-mubtada') is what the sentence is talking about. It could be a noun (the boy; Cairo; Ahmed), pronoun (I; he; they), or noun phrase (the math teacher; a long book).

The predicate (الخبر al-xabar) tells us something about the subject. The Arabic خبر means a piece of news, so you can think of the predicate as delivering news about the subject. It may be an adjective (happy), verb (woke up), or noun (student).

Here are some simple examples. The subject is in bold, and the predicate is underlined.

الولدمصري. (al-walad miSri)

The boyis Egyptian. - The subject is a noun, and the predicate is an adjective.

بنت عميمدرسة. (bint 3ammi mudarrisa)

My cousinis a teacher. - The subject is a noun phrase (a genitive construction), and the predicate is a noun.

هوطويل. (howwa Tawiil)

Heis tall. - The subject is a pronoun, and the predicate is an adjective.

خرجتمنالمبكرا. (xarajat Manaal mubakkiran)

Manalwent out early. - The subject is a (proper) noun, and the predicate is a verb paired with an adverb.

Note: In sentences with a verb, standard Arabic usually follows a Verb-Subject-Object order, though sometimes it uses a Subject-Verb-Object order. The Egyptian dialect pretty much always follows a Subject-Verb-Object order.

Manal went out early.

خرجت منال مبكرا (xarajat Manaal mubakkiran)

منال خرجت بدري (Manaal xargit badri)

The boy ate an apple.

أكل الولد تفاحة (akala l-waladu tuffaaHatan)

الولد كل تفاحة (il-walad kal tuffaaHa)

Negation

Now, how do we negate these sentences? Negating a sentence with a verb is simple; you just negate the verb. (To learn how to negate a verb in Egyptian Arabic, go here.) Negating a sentence without a verb, however, requires a different — but still quite simple — approach. In standard Arabic, you simply insert ليس (laysa), conjugated to match the noun. Here are the conjugations of ليس:

ليس (to not be)

English

Standard Arabic

Singular

I am not

لستُ (lastu)

you (masc.) are not

لستَ (lasta)

you (fem.) are not

لستِ (lasti)

he is not

ليس (laysa)

she is not

ليست (laysat)

Dual

we are not

لسنا (lasna)

you are not

لستما (lastuma)

they (masc.) are not

ليسا (laysaa)

they (fem.) are not

ليستا (laysataa)

Plural

we are not

لسنا (lasna)

you (masc.) are not

لستم (lastum)

you (fem.) are not

لستن (lastunna)

they (masc.) are not

ليسوا (laysuu)

they (fem.) are not

لسن (lasna)

Here are examples of negated verbless sentences in standard Arabic:

أنا لبنانية (ana lubnaaniyya)

أنا لست لبنانية (ana lastu lubnaaniyya)

I am Lebanese.

I am not Lebanese.

بنت عمي مدرسة (bint 3ammi mudarrisa)

بنت عمي ليست مدرسة (bint 3ammi laysat mudarrisa)

My cousin is a teacher.

My cousin is not a teacher.

هو طويل (howwa Tawiil)

هو ليس طويل (howwa laysa Tawiil)

He is tall.

He is not tall.

من اللازم أن تصحو مبكرا (min al-laazim an taSHu mubakkiran)

ليس من اللازم أن تصحو مبكرا (laysa min al-laazim an taSHu mubakkiran)

You need to wake up early.

You don't need to wake up early.

هناك خيار آخر (honaaka xiyaar aaxar)

ليس هناك خيار آخر (laysa honaaka xiyaar aaxar)

There is another choice.

There is no other choice.

لديها خبرة في هذا المجال (ladayha xibra fi haada l-majaal)

ليس لديها خبرة في هذا المجال (laysa ladayha xibra fi haada l-majaal)

She has experience in this field.

She doesn't have experience in this field.

To negate verbless sentences in Egyptian Arabic, you insert مش (miš) between the subject and predicate.

أنا لبنانية (ana libnaniyya)

أنا مش لبنانية (ana miš libnaniyya)

I am Lebanese.

I am not Lebanese.

بنت عمي مدرسة (bint 3ammi mudarrisa)

بنت عمي مش مدرسة (bint 3ammi miš mudarrisa)

My cousin is a teacher.

My cousin is not a teacher.

هو طويل (howwa Tawiil)

هو مش طويل (howwa miš Tawiil)

He is tall.

He is not tall.

لازم تصحى بدري (laazim tiSHa badri)

مش لازم تصحى بدري (miš laazim tiSHa badri)

You need to wake up early.

You don't need to wake up early.

في خيار تاني (fii xiyaar taani)

مافيش خيار تاني (mafiiš xiyaar taani)*

There is another choice.

There is no other choice.

عندها خبرة في المجال ده(3andaha xibra fil-magaal da)

ماعندهاش خبرة في المجال ده (ma3andahaaš xibra fil-magaal da)*

She has experience in this field.

She doesn't have experience in this field.

* Note: In Egyptian Arabic, if a verbless sentence contains an inverted predicate (that is, where the predicate comes before the subject instead of after) consisting of either في, "there is," or a preposition and pronoun suffix, like عندها, then the predicate is negated with the ما...ـش (ma...š) form.